The document describes a method called "Legolization" for automatically generating stable LEGO brick layouts and assemblies from 3D models. It presents a force-based stability analysis that calculates stability metrics for LEGO structures. The method uses an iterative refinement process to optimize layouts by reconfiguring bricks based on stability. It was tested on various models and able to generate assemblies with thousands of bricks that were verified to be physically stable when built. The method accounts for factors like brick limits and external forces. It aims to minimize brick usage while satisfying constraints like shape, color and stability.

1. Sheng-Jie Luo
Yonghao Yue
Chun-Kai Huang
Yu-Huan Chung
Sei Imai
Tomoyuki Nishita
Bing-Yu Chen
National Taiwan University
Columbia University
National Taiwan University
National Taiwan University
The University of Tokyo
UEI Research / Hiroshima Shudo University
National Taiwan University / UEI Research
Legolization
Optimizing LEGO Designs

2. LEGO Construction Activity
Courtesy of Robin Sather - Brickville DesignWorks - LEGO Certified Professional Builder

3. LEGO Sculpture
Courtesy of the LEGO club of The University of Tokyo

4. LEGO Sculpture
Courtesy of Robin Sather - Brickville DesignWorks - LEGO Certified Professional Builder

5. LEGO Construction Problem
© SEGA

6. Legolization
requirements: shape

7. Legolization
requirements: color

8. Legolization
requirements: stability

9. Applications of LEGO Construction
Entertainment/Professional Sculpture
Courtesy of Robin Sather - Brickville DesignWorks - LEGO Certified Professional Builder

10. Applications of LEGO Construction
Rapid prototyping
[Mueller et al. 2014]

11. Applications of LEGO Construction
Architecture designs

12. Previous Approaches
[van Zijl and Smal 2008]
[Gower et al. 1998] [Petrovic 2001]
[Testuz et al. 2013]

13. Previous Approaches
Articulation pointConnectivity Alignment
Heuristic-based metrics:
...

14. Problem of Heuristic-based Metrics
Metric of [Gower et al. 1998]
<1.20 x 104 2.10 x 104

15. Problem of Heuristic-based Metrics
Metric of [Gower et al. 1998]
Metric of [van Zijl and Smal 2008]
Metric of [Petrovic 2001]
Metric of [Testuz et al. 2013]
< 7.30 x 106 2.74 x 107
1.23 x 107 3.13 x 107 1 19
1.20 x 104 2.10 x 104 <
< <

16. Problem Definition

17. Problem Definition
Surface voxel
Inner voxel

18. Problem Definition
Goal: maximal layout

19. Problem Definition
Goal: physically realizable

20. Brick Family
1x1
1x2
1x3
1x4
1x6
1x8
2x2
2x3
2x4
2x6
2x8

21. Fundamental Observation
Per layer:
102 maximal layouts
Total:
1023 maximal layouts

22. Fundamental Observation
Deterministically enumerate all candidates
and check them one by one.
Randomly generate a layout and hope it will
be satisfactory.

23. Core Idea: Force-based Stability Analysis
Structure A Structure B
Stability metric (A) < Stability metric (B)
0 < Stability metric (B)

24. Core Idea: Stability Aware Refinement
Stability
Iteration
Stability
threshold
Bad structures
Good structures
. . .

25. Layout Initialization
Metric: #components
#components = 2

26. Stability-aware Refinement
Structure analysis
Layout reconfiguration
Metric
Portion to
refine

27. Seed brick (structural-critical)
k-ring neighbors (k=1) k-ring neighbors (k=2)
Layout Reconfiguration
Layout
...

28. Layout Reconfiguration
The value of k:
(1) Want enough chance to visit possible layouts
(2) Close neighborhood are usually sufficient
Current fail count
Constant

29. Layout Reconfiguration
Region to refine

30. Brick Operations
(1) Split operation:

31. Brick Operations
(2) Repeated remerge operation:

32. Stability-aware Refinement
Structure analysis
Layout reconfiguration
Metric
Portion to
refine

33. Force-based Stability Analysis
Gravity

34. When is a layout stable?
At each brick:
1. Forces are balanced
2. Forces are within ranges
Gravity

35. Snapping between Bricks

36. Forces Working on Bricks

37. Forces Working on Bricks

38. Forces Working on Bricks

39. Gravity Forces
mg
mg mg

40. Friction Forces

41. Friction Forces

42. Friction Forces

43. Support Forces

44. Support Forces

45. Normal Forces

46. Normal Forces

47. Normal Forces

48. Internal Forces between Bricks
mg
mg mg

49. Force Balance
Translational equilibrium for brick :
mg

50. Force Balance
Rotational equilibrium for brick :
mg

51. One-sided-ness of Forces
Non-negativity condition:

52. Maximum Friction Load
Fi <T Þ Bricks remain snapped

53. Hypothetically,
We could try to solve for
This does not give an answer for unstable cases
T > Fi Î Ff

54. Capacities
For a friction force , define its capacity as

55. Capacities
The capacity for a sculpture

56. Stability Analysis
If forces can be redistributed to make ,
the LEGO sculpture is stable
Stable Unstable
++ -

57. Stability Analysis as an Optimization

58. Stability Analysis as an Optimization
This problem is solvable for both stable and unstable cases

59. Maximum Capacity as Stability Metric
Unstable

60. Threshold for Stability
Stable Unstable

61. Weakest Portion
Unstable

62. Results: Verification of Stability Analysis

63. Results: Verification of Stability Analysis

64. Results: Verification of Stability Analysis

65. Results: Performance
Model #Bricks
Layout Init.
Time
Layout Opt.
Time
Assembly
Time
1642 1.543s 17.89s 8h
1706 0.709s 2054s 9h
8277 67.57s 1259s 4d

66. Results: Performance
Model #Bricks
Layout Init.
Time
Layout Opt.
Time
Assembly
Time
2322 2.347s 16.98s 10h
17755 225.3s 266.1s -
9543 63.09s 108.4s -

67. Results: Accounting for Brick Number Limit

68. Results: Accounting for External Weights
Input external force
500g
Without external force
With external force

69. Results: Building Various LEGO Sculptures
Input voxelized
representation Real assembled sculpture

70. Results: Building Various LEGO Sculptures
Input voxelized
representation
Assembled
sculpture
Refinement
process
#iterations
-50
-40
-30
-20
-10
0
10
20
0 1 2 3

71. Results: Building Various LEGO Sculptures
Input voxelized
representation
Assembled
sculpture
Refinement
process
#iterations
-4
-3
-2
-1
0
1
0 20 40

72. Results: Building Various LEGO Sculptures
Layer 1

73. Results: Building Various LEGO Sculptures
Layer 2

74. Results: Building Various LEGO Sculptures
Layer 2

75. Results: Building Various LEGO Sculptures
…

76. Results: Building Various LEGO Sculptures
Layer 49

77. Results: Building Various LEGO Sculptures
Layer 50

78. Results: Building Various LEGO Sculptures

79. Results: Building Various LEGO Sculptures

80. Limitations
Intermediate assembly can still be unstable
-11.49414.323

81. Limitations
-35
-30
-25
-20
-15
-10
-5
0
5
10
0 50 100 150 200 250 300 350
#iterations
There is no stable layout

82. Conclusions
Stability analysis for LEGO sculptures
Force-based stability metric
Weakest portion

83. Conclusions
Automatically generates LEGO brick layouts
Reducing the number of bricks
Satisfies color constraints
Being able to physical stable

84. Acknowledgements
Christopher Batty, Gabriel Cirio, Anne Fleming,
Eitan Grinspun, Wei-Ting Lin, Han-Wei Liao
Student LEGO Builders from National Taiwan University:
Chin-Yu Chien, Chi-Hao Hsieh, Tsung-Hung Wu, Kang Jao,
Jia-Yu Tsai, Che-Chun Hsu, Fan Wang, Wei-Tse Lee, Yung-Ta Lin,
Li-Ming Yang, Long-Fei Lin, Xiao-Feng Jian, Kai-Han Chang,
Ming-Shiuan Chen
LEGO Certified Professional Builder - Robin Sather
The LEGO Club of The University of Tokyo
Intel, JSPS, MOST

85. Legolization: Optimizing LEGO Designs
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